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Mathematical Modeling with Differential Equations
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Mathematical Modeling with Differential Equations

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (31 March 2023) | Viewed by 9943

Special Issue Editor

Dr. Ioannis Dassios

E-Mail Website
Guest Editor
School of Electrical and Electronic Engineering, University College Dublin, D04 V1W8 Dublin, Ireland
Interests: differential/difference equations; dynamical systems; modeling and stability analysis of electric power systems; mathematics of networks; fractional calculus; mathematical modeling (power systems, materials science, energy, macroeconomics, social media, etc.); optimization for the analysis of large-scale data sets; fluid mechanics; discrete calculus; Bayes control; e-learning
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue aims to collect the latest results on differential/difference equations that can bring new insights in the mathematical modeling of several real-world applications including electrical power systems, materials, energy, macroeconomics, etc. In addition, articles that propose and construct new mathematical models by using differential equations are also very welcome.

This Special Issue will accept high-quality papers with original research results. Its purpose is to bring together mathematicians with physicists, engineers, as well as other scientists.

Topics covered include, but are not limited to:

  • Differential/difference equations
  • Partial differential equations
  • Dynamical systems
  • Control Systems
  • Mathematical modeling
  • Computational modeling and simulation
  • Fractional calculus.

Dr. Ioannis Dassios
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (6 papers)

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Research

19 pages, 5356 KiB  
Article
Three-Temperature Boundary Element Modeling of Ultrasound Wave Propagation in Anisotropic Viscoelastic Porous Media
by Mohamed Abdelsabour Fahmy, Mohammed O. Alsulami and Ahmed E. Abouelregal
Axioms 2023, 12(5), 473; https://doi.org/10.3390/axioms12050473 - 13 May 2023
Cited by 2 | Viewed by 914
Abstract
The main goal of this work is to develop a novel boundary element method (BEM) model for analyzing ultrasonic wave propagation in three-temperature anisotropic viscoelastic porous media. Due to the problems of the strong nonlinearity of ultrasonic wave propagation in three-temperature porous media, [...] Read more.
The main goal of this work is to develop a novel boundary element method (BEM) model for analyzing ultrasonic wave propagation in three-temperature anisotropic viscoelastic porous media. Due to the problems of the strong nonlinearity of ultrasonic wave propagation in three-temperature porous media, the analytical or numerical solutions to the problems under consideration are always challenging, necessitating the development of new computational techniques. As a result, we use a new BEM model to solve such problems. A time-stepping procedure based on the linear multistep method is obtained after solving the discretized boundary integral equation with the quadrature rule. The calculation of a double integral is required to obtain fundamental solutions, but this increases the total BEM computation time. Our proposed BEM technique is used to solve the current problem and improve the formulation efficiency. The numerical results are graphed to demonstrate the effects of viscosity and anisotropy on the nonlinear ultrasonic stress waves in three-temperature porous media. The validity, accuracy, and efficiency of the proposed methodology are demonstrated by comparing the obtained results to a corresponding solution obtained from the finite difference method (FDM). Full article
(This article belongs to the Special Issue Mathematical Modeling with Differential Equations)
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14 pages, 1643 KiB  
Article
Stability Switching in Lotka-Volterra and Ricker-Type Predator-Prey Systems with Arbitrary Step Size
by Shamika Kekulthotuwage Don, Kevin Burrage, Kate J. Helmstedt and Pamela M. Burrage
Axioms 2023, 12(4), 390; https://doi.org/10.3390/axioms12040390 - 17 Apr 2023
Cited by 1 | Viewed by 1198
Abstract
Dynamical properties of numerically approximated discrete systems may become inconsistent with those of the corresponding continuous-time system. We present a qualitative analysis of the dynamical properties of two-species Lotka-Volterra and Ricker-type predator-prey systems under discrete and continuous settings. By creating an arbitrary time [...] Read more.
Dynamical properties of numerically approximated discrete systems may become inconsistent with those of the corresponding continuous-time system. We present a qualitative analysis of the dynamical properties of two-species Lotka-Volterra and Ricker-type predator-prey systems under discrete and continuous settings. By creating an arbitrary time discretisation, we obtain stability conditions that preserve the characteristics of continuous-time models and their numerically approximated systems. Here, we show that even small changes to some of the model parameters may alter the system dynamics unless an appropriate time discretisation is chosen to return similar dynamical behaviour to what is observed in the corresponding continuous-time system. We also found similar dynamical properties of the Ricker-type predator-prey systems under certain conditions. Our results demonstrate the need for preliminary analysis to identify which dynamical properties of approximated discretised systems agree or disagree with the corresponding continuous-time systems. Full article
(This article belongs to the Special Issue Mathematical Modeling with Differential Equations)
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12 pages, 1020 KiB  
Article
Solving Time-Fractional Partial Differential Equation Using Chebyshev Cardinal Functions
by Haifa Bin Jebreen and Carlo Cattani
Axioms 2022, 11(11), 642; https://doi.org/10.3390/axioms11110642 - 14 Nov 2022
Cited by 1 | Viewed by 1657
Abstract
We propose a numerical scheme based on the Galerkin method for solving the time-fractional partial differential equations. To this end, after introducing the Chebyshev cardinal functions (CCFs), using the relation between fractional integral and derivative, we represent the Caputo fractional derivative based on [...] Read more.
We propose a numerical scheme based on the Galerkin method for solving the time-fractional partial differential equations. To this end, after introducing the Chebyshev cardinal functions (CCFs), using the relation between fractional integral and derivative, we represent the Caputo fractional derivative based on these bases and obtain an operational matrix. Applying the Galerkin method and using the operational matrix for the Caputo fractional derivative, the desired equation reduces to a system of linear algebraic equations. By solving this system, the unknown solution is obtained. The convergence analysis for this method is investigated, and some numerical simulations show the accuracy and ability of the technique. Full article
(This article belongs to the Special Issue Mathematical Modeling with Differential Equations)
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8 pages, 331 KiB  
Article
The General Analytic Expression of a Harvested Logistic Model with Slowly Varying Coefficients
by Fahad M. Alharbi
Axioms 2022, 11(11), 585; https://doi.org/10.3390/axioms11110585 - 24 Oct 2022
Cited by 2 | Viewed by 1038
Abstract
The harvested logistic model with a slow variation in coefficients has been considered. Two cases, which depend on the harvest rate, were identified. The first one is when the harvest is subcritical, where the population evolves to an equilibrium. The other is supercritical [...] Read more.
The harvested logistic model with a slow variation in coefficients has been considered. Two cases, which depend on the harvest rate, were identified. The first one is when the harvest is subcritical, where the population evolves to an equilibrium. The other is supercritical harvesting, where the population decreases to zero at finite times. The single analytic approximate expression, which is capable of describing both harvesting cases, is readily and explicitly obtained using the multi-time scaling method together with the perturbation approach. This solution fits for a wide range of coefficient values. In addition, such an expression is validated by utilizing numerical computations, which are obtained by using the fourth-order Runge–Kutta technique. Finally, the comparison shows a very good agreement between the two methods. Full article
(This article belongs to the Special Issue Mathematical Modeling with Differential Equations)
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15 pages, 1216 KiB  
Article
A Reliable Technique for Solving Fractional Partial Differential Equation
by Azzh Saad Alshehry, Rasool Shah, Nehad Ali Shah and Ioannis Dassios
Axioms 2022, 11(10), 574; https://doi.org/10.3390/axioms11100574 - 20 Oct 2022
Cited by 13 | Viewed by 1778
Abstract
The development of numeric-analytic solutions and the construction of fractional-order mathematical models for practical issues are of the greatest importance in a variety of applied mathematics, physics, and engineering problems. The Laplace residual-power-series method (LRPSM), a new and dependable technique for resolving fractional [...] Read more.
The development of numeric-analytic solutions and the construction of fractional-order mathematical models for practical issues are of the greatest importance in a variety of applied mathematics, physics, and engineering problems. The Laplace residual-power-series method (LRPSM), a new and dependable technique for resolving fractional partial differential equations, is introduced in this study. The residual-power-series method (RPSM), a well-known technique, and the Laplace transform (LT) are elegantly combined in the suggested technique. This innovative approach computes the fractional derivative in the Caputo sense. The proposed method for handling fractional partial differential equations is provided in detail, along with its implementation. The novel approach yields a series solution to fractional partial differential equations. To validate the simplicity, effectiveness, and viability of the suggested technique, the provided model is tested and simulated. A numerical and graphical description of the effects of the fractional order γ on approximating the solutions is provided. Comparative results show that the suggested method approximates more precisely than current methods such as the natural homotopy perturbation method. The study showed that the aforementioned method is straightforward, trustworthy, and suitable for analysing non-linear engineering and physical issues. Full article
(This article belongs to the Special Issue Mathematical Modeling with Differential Equations)
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22 pages, 5534 KiB  
Article
A Demand Side Management Control Strategy Using RUNge Kutta Optimizer (RUN)
by Ankit Kumar Sharma, Ahmad M. Alshamrani, Khalid A. Alnowibet, Adel F. Alrasheedi, Akash Saxena and Ali Wagdy Mohamed
Axioms 2022, 11(10), 538; https://doi.org/10.3390/axioms11100538 - 08 Oct 2022
Viewed by 2141
Abstract
Demand side management initiatives have gained attention recently because of the development of the smart grid and consumer-focused regulations. The demand side management programme has numerous goals. One of the main goals is to control energy demand by altering customer demand. This can [...] Read more.
Demand side management initiatives have gained attention recently because of the development of the smart grid and consumer-focused regulations. The demand side management programme has numerous goals. One of the main goals is to control energy demand by altering customer demand. This can be done in several ways, including financial discounts and behaviour changes brought about by providing knowledge to support the grid’s stressed conditions. In this study, demand side management techniques for future smart grids are presented, including load shifting and strategic conservation. There are many controlled devices on the grid. The load shifting and day before strategic conservation approaches mentioned in this study are derived analytically for the minimization problem. For resolving this minimization issue, the RUNge Kutta optimizer (RUN) was developed. On a test smart grid with two service zones, one with residential consumers and the other with commercial consumers, simulations are performed. By contrasting the outcomes with the slime mould algorithm (SMA), Sine Cosine Algorithm (SCA), moth–flame optimization (MFO), and whale optimization algorithm (WOA), RUN demonstrates its effectiveness. The simulation findings demonstrate that the suggested demand side management solutions produce significant cost savings while lowering the smart grid’s peak load demand. Full article
(This article belongs to the Special Issue Mathematical Modeling with Differential Equations)
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